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IntroductionRecognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design at the corner, only two different shapes were actually drawn. The design was put together by copying and manipulating these shapes to produce versions of them of different sizes and in different positions.
Similar and Congruent Figures
• Congruent triangles have all sides congruent and all angles congruent.
• Similar triangles have the same shape; they may or may not have the same size.
Note: Two figures can be similar but not congruent, but they can’t be congruent but not similar. Think about why!
Similar and Congruent Figures
Ratios and Similar Figures• Similar figures have corresponding
sides and corresponding angles that are located at the same place on the figures.
• Corresponding sides have to have the same ratios between the two figures.
• A ratio is a comparison between 2 numbers (usually shown as a fraction)
Ratios and Similar Figures
Example
A E
C
F
D
G H
B
These sides correspond:
AB and EF
BD and FH
CD and GH
AC and EG
These angles correspond:
A and E
B and F
D and H
C and G
Ratios and Similar Figures
Example
7 m
3 m 6 m
14 m
These rectangles are similar, because the ratios of these corresponding sides are equal:
7 14
3 6
3 6
7 14
7 3
14 6
14 6
7 3
•A proportion is an equation that states that two ratios are equal.
•Examples:4 8
10n
6
3 2
m
n = 5 m = 4
Proportions and Similar Figures
Proportions and Similar Figures
You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 16
m
10 m
n
5 m
Proportions and Similar Figures
You can use proportions of corresponding sides to figure out unknown lengths of sides of polygons. 16
m
10 m
n
5 m
Similar triangles• Similar triangles are triangles with the same shape
For two similar triangles, • corresponding angles have the same measure
• length of corresponding sides have the same ratio
65o
25o
A4 cm 2cm
12cmB
Example
Angle A = 90o Side B = 6 cm
Proportions and Similar Figures
Can you solve for the missing variable in these similar triangles?
J20
12
12